Sublinear graph-bootstrap running-time conjecture

Let $H$ be a graph, and let $M_H(n)$ denote the maximum running time of the $H$-bootstrap process over all starting graphs on $n$ vertices. Sublinear running-time conjecture. Every graph $H$ with

$$M_H(n)=o(n)$$

satisfies either

$$M_H(n)=O(1)$$

or

$$M_H(n)=\Theta(\log n).$$

A negative answer to the preceding degree-one-vertex question would motivate this dichotomy, asserting that below linear running times only bounded and logarithmic orders occur. The source does not state a resolution of this conjecture.